| L(s) = 1 | + (15.0 − 5.43i)2-s + (31.9 + 77.0i)3-s + (196. − 163. i)4-s + (280. − 676. i)5-s + (899. + 986. i)6-s + (−125. + 125. i)7-s + (2.07e3 − 3.53e3i)8-s + (−282. + 282. i)9-s + (542. − 1.17e4i)10-s + (3.70e3 − 8.94e3i)11-s + (1.88e4 + 9.96e3i)12-s + (1.76e4 + 4.25e4i)13-s + (−1.20e3 + 2.56e3i)14-s + 6.11e4·15-s + (1.20e4 − 6.44e4i)16-s + 6.17e4i·17-s + ⋯ |
| L(s) = 1 | + (0.940 − 0.339i)2-s + (0.394 + 0.951i)3-s + (0.769 − 0.638i)4-s + (0.448 − 1.08i)5-s + (0.693 + 0.761i)6-s + (−0.0521 + 0.0521i)7-s + (0.506 − 0.861i)8-s + (−0.0430 + 0.0430i)9-s + (0.0542 − 1.17i)10-s + (0.253 − 0.611i)11-s + (0.911 + 0.480i)12-s + (0.617 + 1.49i)13-s + (−0.0313 + 0.0667i)14-s + 1.20·15-s + (0.184 − 0.982i)16-s + 0.739i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.414i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.85330 - 0.837231i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.85330 - 0.837231i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-15.0 + 5.43i)T \) |
| good | 3 | \( 1 + (-31.9 - 77.0i)T + (-4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (-280. + 676. i)T + (-2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (125. - 125. i)T - 5.76e6iT^{2} \) |
| 11 | \( 1 + (-3.70e3 + 8.94e3i)T + (-1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (-1.76e4 - 4.25e4i)T + (-5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 - 6.17e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (-7.83e4 + 3.24e4i)T + (1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (3.19e5 + 3.19e5i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 + (9.92e5 - 4.11e5i)T + (3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 + 4.69e4iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (2.43e5 - 5.88e5i)T + (-2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (2.02e6 - 2.02e6i)T - 7.98e12iT^{2} \) |
| 43 | \( 1 + (2.79e5 - 6.75e5i)T + (-8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 + 5.78e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (2.69e6 + 1.11e6i)T + (4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (-3.71e6 - 1.54e6i)T + (1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (-1.07e7 + 4.46e6i)T + (1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (1.07e7 + 2.60e7i)T + (-2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (1.20e7 - 1.20e7i)T - 6.45e14iT^{2} \) |
| 73 | \( 1 + (9.32e6 - 9.32e6i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 - 5.30e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (4.28e7 - 1.77e7i)T + (1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (2.36e7 + 2.36e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 - 1.72e8T + 7.83e15T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.74570880174988186895094332033, −13.76246033581403013088993294865, −12.61294362446804781903018336162, −11.28087212482270671525105678905, −9.818121417665019398649680576722, −8.802616220102303443084039522381, −6.28403396807695169854289257109, −4.75822499588664005605822611580, −3.68293468627486423083915445219, −1.53836458901425759723781753006,
1.97933450991700161684502222711, 3.35098709026827658971345443868, 5.67440073180215086056253882162, 7.00721149701227963041316624142, 7.85277738738421079435061940334, 10.22123576837315678194484894232, 11.70341650394481444517254664764, 13.05538956164956614766531438874, 13.77306307140535115911649295678, 14.79472883228168015097154250109
